Optimization and Engineering

, Volume 20, Issue 1, pp 65–88 | Cite as

Differential evolution with the adaptive penalty method for structural multi-objective optimization

  • Dênis E. C. VargasEmail author
  • Afonso C. C. Lemonge
  • Helio J. C. Barbosa
  • Heder S. Bernardino
Research Article


Real-world engineering design problems, like structural optimization, can be characterized as a multi-objective optimization when two or more conflicting objectives are in the problem formulation. The differential evolution (DE) algorithm is nowadays one of the most popular meta-heuristics to solve optimization problems in continuous search spaces and has attracted much attention in multi-objective optimization due to its simple implementation and efficiency when solving real-world problems. A recent paper has shown that GDE3, a well-known DE-based algorithm, performs efficiently when solving structural multi-objective optimization problems. Also an adaptive penalty technique called APM was adopted to handle constraints. However, the authors did not investigate the contribution of this technique and that of the GDE3 algorithm separately. So, in this work, the results obtained by GDE3 equipped with the APM scheme (denoted here by GDE3 + APM) are compared with those found by the original GDE3 in order to investigate the advantages and limitations of this constraint handling technique in those problems. The results of the GDE3 + APM are also compared with the most commonly used multi-objective meta-heuristic, namely NSGA-II, in order to comparatively evaluate the quality of the solutions obtained with respect to other algorithms from the literature. The analysis indicates that GDE3 + APM is more efficient than both GDE3 and NSGA-II in most performance metrics used when solving the structural multi-objective optimization problems considered here, suggesting that the GDE3 + APM algorithm is promising in this area, and that the APM technique makes a considerable contribution to its performance.


Structural multi-objective optimization Differential evolution Constraint handling Adaptive penalty method 



The authors thank CNPq (Grants 310778/2013-1 and 306186/2017-9) and FAPEMIG (Grants TEC PPM 528/11, TEC PPM 388/14, and APQ-03414-15).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Dênis E. C. Vargas
    • 1
    Email author
  • Afonso C. C. Lemonge
    • 2
  • Helio J. C. Barbosa
    • 2
    • 3
  • Heder S. Bernardino
    • 2
  1. 1.Instituto Federal de EducaçãoCiência e Tecnologia do Sudeste de Minas GeraisRio PombaBrazil
  2. 2.Universidade Federal de Juiz de ForaJuiz de ForaBrazil
  3. 3.Laboratório Nacional de Computação CientíficaPetrópolisBrazil

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